let $r$ be a commutative ring and $m$ be an $r$-module with $t(m)$ as subset,   the set of torsion elements. the total graph of the module denoted   by $t(gamma(m))$, is the (undirected) graph with all elements of   $m$ as vertices, and for distinct elements  $n,m in m$, the   vertices $n$ and $m$ are adjacent if and only if $n+m in t(m)$. in this paper we   study the domination number of $t(ga...

In this paper we study the signed Roman dominationnumber of the join of graphs. Specially, we determine it for thejoin of cycles, wheels, fans and friendship graphs.

Let G be a graph with vertex set V ( ). A total Italian dominating function (TIDF) on is f : ) → {0, 1, 2} such that (i) every v = 0 adjacent to u 2 or two vertices w and z (ii) ≥ 1 1. The domination number γ tI the minimum weight of function. In this paper, we present Nordhaus–Gaddum type inequalities for number.

The paired bondage number (total restrained bondage number, independent bondage number, k-rainbow bondage number) of a graph G, is the minimum number of edges whose removal from G results in a graph with larger paired domination number (respectively, total restrained domination number, independent domination number, k-rainbow domination number). In this paper we show that the decision problems ...

For a positive integer k, a k-subdominating function of a graph G=(V; E) is a function f :V →{−1; 1} such that ∑u∈NG [v] f(u)¿ 1 for at least k vertices v of G. The ksubdomination number of G, denoted by ks(G), is the minimum of ∑ v∈V f(v) taken over all k-subdominating functions f of G. In this article, we prove a conjecture for k-subdomination on trees proposed by Cockayne and Mynhardt. We al...

A {em Roman dominating function} on a graph $G$ is a function $f:V(G)rightarrow {0,1,2}$ satisfying the condition that every vertex $u$ for which $f(u)=0$ is adjacent to at least one vertex $v$ for which $f(v)=2$. A {em total Roman dominating function} is a Roman dominating function with the additional property that the subgraph of $G$ induced by the set of all vertices of positive weight has n...

a set $s$ of vertices in a graph $g=(v,e)$ is called a total$k$-distance dominating set if every vertex in $v$ is withindistance $k$ of a vertex in $s$. a graph $g$ is total $k$-distancedomination-critical if $gamma_{t}^{k} (g - x) < gamma_{t}^{k}(g)$ for any vertex $xin v(g)$. in this paper,we investigate some results on total $k$-distance domination-critical of graphs.

Let G = (V , E) be a simple graph on vertex set V and define a function f : V → {−1,1}. The function f is a signed dominating function if for every vertex x ∈ V , the closed neighborhood of x contains more vertices with function value 1 than with −1. The signed domination number of G, γs(G), is the minimum weight of a signed dominating function on G. We give a sharp lower bound on the signed do...